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%I #69 Oct 14 2022 07:02:49
%S 1,1,1,2,2,1,5,5,4,2,16,16,14,10,5,61,61,56,46,32,16,272,272,256,224,
%T 178,122,61,1385,1385,1324,1202,1024,800,544,272,7936,7936,7664,7120,
%U 6320,5296,4094,2770,1385,50521,50521,49136,46366,42272,36976,30656,23536,15872,7936,353792
%N Triangle of Euler-Bernoulli or Entringer numbers.
%C T(n, k) is the number of up-down permutations of n starting with k where 1 <= k <= n. - _Michael Somos_, Jan 20 2020
%D R. C. Entringer, A combinatorial interpretation of the Euler and Bernoulli numbers, Nieuw Archief voor Wiskunde, 14 (1966), 241-246.
%H Alois P. Heinz, <a href="/A010094/b010094.txt">Rows n = 1..150, flattened</a> (first 51 rows from Vincenzo Librandi)
%H B. Bauslaugh and F. Ruskey, <a href="https://doi.org/10.1007/BF01932127">Generating alternating permutations lexicographically</a>, Nordisk Tidskr. Informationsbehandling (BIT) 30 16-26 1990.
%H D. Foata and G.-N. Han, <a href="http://arxiv.org/abs/1304.2485">Secant Tree Calculus</a>, arXiv preprint arXiv:1304.2485 [math.CO], 2013.
%H Dominique Foata and Guo-Niu Han, <a href="http://www-irma.u-strasbg.fr/~foata/paper/pub123Seidel.pdf">Seidel Triangle Sequences and Bi-Entringer Numbers</a>, November 20, 2013.
%H Foata, Dominique; Han, Guo-Niu; Strehl, Volker <a href="https://doi.org/10.1016/j.laa.2016.09.016">The Entringer-Poupard matrix sequence</a>. Linear Algebra Appl. 512, 71-96 (2017). example 4.4
%H M. Josuat-Verges, J.-C. Novelli and J.-Y. Thibon, <a href="http://arxiv.org/abs/1110.5272">The algebraic combinatorics of snakes</a>, arXiv preprint arXiv:1110.5272 [math.CO], 2011.
%H J. Millar, N. J. A. Sloane and N. E. Young, A new operation on sequences: the Boustrophedon transform, J. Combin. Theory, 17A (1996) 44-54 (<a href="http://neilsloane.com/doc/bous.txt">Abstract</a>, <a href="http://neilsloane.com/doc/bous.pdf">pdf</a>, <a href="http://neilsloane.com/doc/bous.ps">ps</a>).
%H C. Poupard, <a href="https://doi.org/10.1016/0012-365X(82)90293-X">De nouvelles significations énumeratives des nombres d'Entringer</a>, Discrete Math., 38 (1982), 265-271.
%F T(1, 1) = 1; T(n, n) = 0 if n > 1; T(n, k) = T(n, k+1) + T(n-1, n-k) if 1 <= k < n. - _Michael Somos_, Jan 20 2020
%e From _Vincenzo Librandi_, Aug 13 2013: (Start)
%e Triangle begins:
%e 1;
%e 1, 1;
%e 2, 2, 1;
%e 5, 5, 4, 2;
%e 16, 16, 14, 10, 5;
%e 61, 61, 56, 46, 32, 16;
%e 272, 272, 256, 224, 178, 122, 61;
%e 1385, 1385, 1324, 1202, 1024, 800, 544, 272;
%e 7936, 7936, 7664, 7120, 6320, 5296, 4094, 2770, 1385;
%e ... (End)
%e Up-down permutations for n = 4 are k = 1: 1324, 1423; k = 2: 2314, 2413; k = 3: 3411; k = 4: none. - _Michael Somos_, Jan 20 2020
%p b:= proc(u, o) option remember; `if`(u+o=0, 1,
%p add(b(o-1+j, u-j), j=1..u))
%p end:
%p T:= (n, k)-> b(n-k+1, k-1):
%p seq(seq(T(n, k), k=1..n), n=1..12); # _Alois P. Heinz_, Jun 03 2020
%t e[0, 0] = 1; e[_, 0] = 0; e[n_, k_] := e[n, k] = e[n, k-1] + e[n-1, n-k]; Join[{1}, Table[e[n, k], {n, 0, 11}, {k, n, 1, -1}] // Flatten] (* _Jean-François Alcover_, Aug 13 2013 *)
%o (PARI) {T(n, k) = if( n < 1 || k >= n, k == 1 && n == 1, T(n, k+1) + T(n-1, n-k))}; /* _Michael Somos_, Jan 20 2020 */
%Y Columns k=1,3-4 give: A000111, A006212, A006213.
%Y Row sums give A000111(n+1).
%Y Cf. A008282.
%K nonn,tabl,easy,nice
%O 1,4
%A _N. J. A. Sloane_
%E More terms from Will Root (crosswind(AT)bright.net), Oct 08 2001
%E Irregular zeroth row deleted by _N. J. A. Sloane_, Jun 04 2020
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